Complex Numbers and Complex Algebra: Taylors Series
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The following theorem shows that the power series represents an analytic function in the domain of convergence.
Theorem: Let the series MATH have radius of convergence $R > 0$. Then,

    a.
  1. The function defined by MATH is analytic in MATH.
  2. For each $k \ge 1$, the series MATH has the radius of converges $R$.
  3. If $f^{(k)}(z)$ is the $k$-th derivative of $f(z)$ then MATH
  4. For MATH, the coefficient MATH.

Example: The power series MATH has the radius of convergence $R = \infty$ and hence the function MATH is analytic in $\QTR{Bbb}{C}$. Further, observe that MATH and this series also has the radius of convergence $R = \infty$.

   
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